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研究生： 洪雅婷
Ya-Ting Hung
論文名稱：
On the Diophantine Equation of (x^m-1)/(x-1)=(y^n-1)/(y-1)
指導教授： 陳燕美
Yen-Mei J. Chen
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
畢業學年度： 94
語文別： 英文
論文頁數： 25
外文關鍵詞： Diophantine Equation
相關次數： 點閱：10  下載：0 
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    • 我們考慮特別的整係數方程式去尋找整數解或有理數解。Ratat和Goormaghtigh觀察出當x,y,m,n為正整數時，(x,y,m,n)=(5,2,3,5)和(90,2,3,13)是方程式 (x^m-1)/(x-1)=(y^n-1)/(y-1) 的解。因此，猜想此方程式只有這兩組解。現在，我們集中焦點在m=3。此時方程式有兩組已知的解。除了那兩組解之外的解就稱為例外解。這篇論文，主要是考慮當n=4時，此方程式沒有例外解。

    We consider special Diophantine equations with integral coefficient and seek
    integral or rational solutions. Ratat[1] and Goormaghtigh [2] observed that
    31=(2^5-1)/(2-1)=(5^3-1)/(5-1)
    and 8191=(2^13-1)/(2-1)=(90^3-1)/(90-1)
    are solutions of the Diophantine equation
    (x^m-1)/(x-1)=(y^n-1)/(y-1)
    ; x > 1; y > 1; n > m > 2.....(1)
    Now, we will focus our attention on the equation
    (x^3-1)/(x-1)=(y^n-1)/(y-1)
    ; n > 2; x > 1; y > 1 with x > y.....(2)
    Equation (2) has two known solutions (x, y, n) = (5, 2, 5), (90, 2, 13). Any other
    solution (x, y, n) of (2) will be called exceptional. In this paper, we show that this
    equation (2) has no exceptional solution when n = 4.

    Contents Abstract ........................................................................................ii Outlines .......................................................................................iii 0 Introduction .................................................................................1 1 The Diophantine Equation (x^3-1)/(x-1)=(y^n-1)/(y-1).........................3 2 The Method of Descent .....................................................................6 3 The Diophantine Equation (x^3-1)/(x-1)=(y^4-1)/(y-1).....................12 References......................................................................................25

    References
    [1] R. Ratat L''intermediaire des Mathematiciens, 23 (1916),150.
    [2] R. Goormaghtigh, L''intermediaire des Mathematiciens, 24 (1917), 88.
    [3] A. Makowski, A. Schinzel, Sur I"equation indet''ermin''ee de R. Goormaghtigh, Mathesis 68 (1959), 128-142.
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    [5] T. Nagell, The diophantine equation x^2 + 7 = 2^n, Ark. Mat. 4 (1961) 185-187.
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    [10] G. Bergmann, U ber Eulers Beweis des grossen Fermatschen Satzes fur den Exponenten 3:, Math. Ann., 164(1996), 159-175.
    [11] Maohua Le, Exceptional solutions of the exponential diophantine equation (x^3-1)/(x-1)=(y^n-1)/(y-1), J. reine angew. Math. 543 (2002), 187-192.
    [12] Yu. V. Nesterenko and T. N. Shorey, On an equation of Goormaghtigh, Acta Arith. 83 (1998), 381-389.
    [13] Pingzhi Yuan, On the diophantine equation (x^3-1)/(x-1)=(y^n-1)/(y-1), Journal of Number Theory 112 (2005), 20-25.
    [14] Kenneth Ireland, Michael Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag New York Inc., 1982.
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    [16] L. J. Mordell, Diophantine Equations, Cambridge, England, 1969.25

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